Band gap and quantum valley Hall interface state in uniaxial strain superlattice of graphene

Abstract

We investigate the band structures of strained monolayer and bilayer graphene superlattice, which is formed by subjecting graphene to a periodic uniaxial strain. The strain superlattice is attained by imposing distinctively positive and negative strains on opposite halves of the supercell. A controllable band gap and partial flat band are observed in superlattice, with the strain applied along the zigzag and armchair direction respectively. The band gap can be achieved with a small strain applied, and the magnitude of band gap can be tuned by adjusting the strength and smoothness of the strain, with maximal band gaps reaching 1200 meV and 900 meV for monolayer and bilayer graphene, respectively. The partial flat band can be used in inducing quantum valley Hall interface state (QVHIS) localized at the strain interface of bilayer strain superlattice, with a vertical electric field applied simultaneously. Our results provide a strategy for creating controllable band gap or QVHIS in graphene, which could be useful in designing graphene-based electronic devices.

Graphical abstract

Keywords

graphene / strain superlattice / band gap / quantum valley Hall edge stste

1 Introduction

Since its successful isolation, graphene has sparked immense interest as a cutting-edge electronic material due to its extraordinary electronic properties, which have fueled both fundamental research and promising applications [1–8]. At the same time, the advancement of electronic devices lies the strategic manipulation of electronic properties through band engineering. In the realm of electronic device design, band engineering plays a pivotal role in effectively manipulating the electronic properties of graphene.

The design of graphene-based electronic devices, whether monolayer or bilayer, presents significant hurdles due to its inherent gapless electronic spectrum [9–13]. To overcome this limitation, researchers have employed various band engineering techniques, such as interactions with substrates [14–17], doping [18–23], quantum confinement [24, 25], and strain manipulation [26–28]. Despite these efforts, challenges persist: substrate and chemical doping can introduce impurities that compromise carrier mobility, and achieving precise control over graphene nanoribbon edges and widths remains a technical obstacle. A substantial uniform uniaxial strain is required to achieve a band gap [28]. Moreover, the application of substantial strain to graphene can potentially undermine its overall performance. There is an urgent need for the development of efficient and reliable techniques that can create a tunable and substantial bandgap in graphene.

The quantum valley Hall edge state (QVHES) hold significant potential for low-dissipation transport channels and topological transistors or devices [29–32]. In graphene nanoribbons, the presence of a partial flat band can give rise to QVHESs in both zigzag [33–38] and shear-deformed armchair structures [39]. However, the sensitivity of QVHES in graphene nanoribbon to the edge perfection poses a challenge for their practical applications. Even though the QVHES has been explored in dual-split-gated bilayer graphene devices [31, 32], practical implementation of the split gate poses significant challenges, and experimental findings indicate that achieving the desired QVHES with the current setup remains a high-bar requirement. In graphene, a key limitation arises when the flat band merges in graphene, restricting the practical application of QVHESs.

Nowadays, strain engineering has been a powerful tool to modified the electric properties of graphene [28, 40, 41]. Graphene is remarkable sensitivity to external strain, and can withstand mechanical deformation of up to 25% [28, 40]. The applied strain induces a gauge pseudo-vector potential to the graphene, which exhibits distinct signs for the two valleys (K and K^{\prime }) in graphene. This property effectively alters the electronic structure of graphene.

In this investigation, we delve into the band structures of strain superlattices of graphene [42–44]. The strain superlattice is synthesized by subjecting the graphene to a periodic uniaxial strain. This is accomplished by applying positive uniaxial strain to one half of the supercell of superlattice, while negative uniaxial strain on the other half, with the smoothness of the strain field variable to emulate diverse strain fields. Notable features emerge: when the periodic strain is applied along the zigzag direction, a band gap can be achieved with small strain. The band gap is tunable by the strain, with the maximal gap in monolayer/bilayer graphene 1200/900 meV. Conversely, in the case of the strain applied along armchair direction, partial flat bands arise, giving rise to quantum valley Hall interface state (QVHIS) in bilayer graphene when subjected to a vertical electric field simultaneously. The QVHIS appears at the interface between regions of positive and negative strain, which is essentially the QVHES attached to the interface.

This work is organized as follows. Section 2 provides an introduction to the structure and model of monolayer and bilayer graphene with a uniaxial strain superlattice or an uniforaml uniaxial strain and external potential superlattice. Section 3 presents the calculation results and their analysis. Finally, in Section 4, the conclusion of this study is presented.

2 Model and structure

The strained monolayer/bilayer graphene superlattice, denoted as MGS/BGS, is formed by applying periodic non-uniform strain along the x-direction, shown in Fig.1(a)/(c) and (b)/(d) respectively. For the convenience of discussion, the configuration in Fig.1(a)/(b)/(c)/(d) is referred to as zMGS/aMGS/zBGS/aBGS. The supercells of all superlattices are encircled by the blue dotted rectangles, as depicted in Fig.1(a)−(d). L_x and L_y are denoted as the length of supercell along x- and y-direction respectively, with L_x=\sqrt{3}(N -1)a for zMGS and zBGS, and L_x=(\frac{3N}{2}-1)a for aMGS and aBGS. a = 0.142 nm is the length of unstrained C−C bond. The Hamiltonian of the superlattices can be written based on the tight-binding as

(1)\begin{array}{rl}H=& \sum _{l⟨ij⟩}{t}_{ij}{a}_{li}^{†}{b}_{lj}+t_{\perp }\sum _i{b}_{2i}^{†}{a}_{1i}\\ & +\sum _{li}(-1{)}^l\frac{V}{2}\left(a_{li}^{†}{a}_{li}+b_{li}^{†}{b}_{li}\right)+H.c.,\end{array}

a_{li}/b_{li} represents the annihilation operator at A_i/B_i-atom of layer l = 1, 2. t_{ij}=texp[-\beta (\frac{r_{ij}}{a} -1)], where t=3\mathrm{e}\mathrm{V} is the nearest hopping energy before deformation. \beta =3.37 represents the the Grüneisen parameter [28]. r_{ij}=|{\mathit{r}}_{\mathit{i}}-{\mathit{r}}_{\mathit{j}}| is the length of the strained bond between atom i and j in the same layer. t_{\perp }=0.1t denotes the interlayer hopping in bilayer graphene [45]. V stands for potential difference induced by applying a vertical electric field to BGS.

When the uniaxial strain is applied, the deformed vectors can be obtained as

(2){\mathit{r}}_{ij}=(I +{\epsilon }^{\prime } ){\mathit{r}}_{\mathbf{0}} ,

where I is the identity matrix, and

(3){\epsilon }^{\prime }= {\epsilon }_{\nu }(x)\left(\begin{array}{cc}1& 0\\ 0& -\sigma \end{array}\right) .

\sigma = 0.165 being the Poisson’s ratio of graphene [28]. In this study, the periodic strain is applied along the x-direction, with {\epsilon }_v(x )={\epsilon }_v(x +L_x). For x\in \left(-\frac{L_x}{2},\frac{L_x}{2}\right], {\epsilon }_v(x) is given as

(4){\epsilon }_v(x)= {\epsilon }_0\mathrm{S}\mathrm{g}\mathrm{n}(x)|\sin {\displaystyle \frac{2\pi x}{L_x}}{|}^{\nu },

to simulate the non-uniform uniaxial strain. {\epsilon }_0 denotes the maximal magnitude of the non-unifrom strain, \mathrm{S}\mathrm{g}\mathrm{n}(x) is the sign function. \nu is chosen from 0 to 1, which defines the smoothness of the strain field.

3 Results and discussion

3.1 Band gap in zMGS and zBGS

After performing calculatins, the band gap (E_g) in zMGS and zBGS is found and exhibits dependence on the width of the supercell (N) and the strain ({\epsilon }_0), as illustrated in Fig.2(a)−(d). The band gap is observed when valley K and K^{\prime } merging either at the Brillouin zone (BZ) center or its boundary. As we know, valley K and K^{\prime } in graphene exhibit an opposite shifting along the armchiar direction of BZ under uniaxial strain [46], regardless of the strain direction being aligned with the zigzag or armchair directions. This phenomenon facilitates the valley matching process.

In detail, the valley motion and matching process are shown in Fig.3. The valley motion is characterized by three distinct mechanisms, depending on the initial location of valley K and K^{\prime }. When the width of supercell N=3n (n\in {\mathbb{N}}^{+}), two Dirac cones folded at the BZ center [46–50]; conversely, for N=3n\pm 1, valley K is located at \pm \frac{2\pi }{3L_x} respectively. This leads to three distinct valley motion patterns, the band gap opens at either the BZ center or its boundaries, which are visually depicted in Fig.3. For N=3n, the band gap open at the center of BZ with the strain increased, then the gap closed again and two valleys shift to the boundary of BZ, shown in Fig.3(a)−(d). For N=3n+ 1/N=3n- 1, both valleys shifting to the boundary/center of BZ when the strain increased, then open a band gap at the boundary/center of BZ, shown in Fig.3(e)−(h)/(i)−(l).

Upon setting N, the band gap can be control by altering strain. For aMGS with small N, a substantial band gap emerges when subjected to a strain exceeding 13\mathrm{\%}. Specifically, for N=4, the band gap (E_g) opens at strains above 13\mathrm{\%}, and with further strain increment, it reaches a maximum of approximately E_g\sim 0.4t (1200 meV). Similar to aMGS, the maximal band gap in N=4 aBGS reaching 0.3t\sim 900 meV. As N increases, we observed that the band gap opened twice, which means that valley K and K^{\prime } matching twice within the strain range investigated. In Fig.2(e)−(h) for N=12, the band gap open at strains ranging from 1% to 14% and approximately 20%, with the maximum value being around E_g\sim 0.06t. Intriguingly, even small strains (\sim 3\mathrm{\%}) could trigger a noticeable gap, shown in Fig.3(e)−(h).

The calculation results also reveal that the band gap oscillates as a function of N, shown in Fig.2(e)−(h). Across the strain range investigated, a total of six distinct band gap peaks are discerned as N is varied. At an applied strain of {\epsilon }_0=10\mathrm{\%}, the maximal E_g could reach approximately 0.04t, which corresponds to approximately 120 meV for zMGS with N=28.

The smoothness of strain field has a substantial impact on the band gap, with lower values of \nu generally resulting in a larger band gap. By comparing results in Fig.2(e) and (f), the maximum band gap decreases from 5.5\times {10}^{-2}t to 1.25\times {10}^{-2}t when \nu changes from 0 to 0.15.

3.2 Partial flat band and QVHIS in aMGS and aBGS

When the strain is applied along the armchair direction, the partial flat bands are observed in aMGS and aBGS, shown in Fig.4. Such periodic strain preserves the translation symmetry along zigzag direction, leading to valley splitting, as the combination of positive and negtive uniaxial strain will split both valley K and K^{\prime } [51]. The length of partial flat bands are proposal to the strength of strain {\epsilon }_0. In Fig.4, we found there are more subbands appeared around valleys that approching to the boundary of BZ at low energy range, as the positive/negtive strain will expand/narrow the supercell width. The charge density of electrons in the partial flat band is depicted in Fig.5, where the charge density is prominently confined to the interfaces between regions of positive and negative strain.

Given the inherent difficulty in realizing staggered potential in monolayer graphene, our study delves into the impact of vertical electric fields in bilayer systems. Similar to aMGS, we observed the presence of two partial flat bands in the band structure of aBGS, shown in Fig.5(c). For N=200, we introduce a potential difference V=0.04t between the upper and bottom layer of graphene. The band structures are indicating at Fig.6(a)−(d). Without strain, a subgap appeared with E_g=0.04 t. When {\epsilon }_0 is changed to 2%, the quantum valley Hall interface states (QVHISs) [33–39] appeared in the subgap, shown in Fig.6(b). Eight QVHISs are found as strain exceeding {\epsilon }_0 = 2%.

The charge densities |\psi {|}^2 of QVHIS by setting the Fermi energy slightly above E=0 are shown in Fig.7(a)−(c). Four QVHISs are found in Fig.6(b)−(d), which are concentrated at the center or one side of supercell, penetrating into the strain regions [Fig.7(a)−(c)]. The penetration depth is narrowed when {\epsilon }_0 increased from 2% to 10%. By comparing Fig.7(b) and (c), we found the charge density barely changes, which means that the QVHISs exhibit similar at the strain range exceeding 10%.

At low energies, eight QVHISs can be effortlessly detected in stain superlattices featuring a wide supercell. These is due to the increasing width suppressing the intervalley scattering, leading the suppression of the interaction gap. For example, upon setting {\epsilon }_0 = 20% and Fermi energy, eight QVHISs are found when N is changed to 400, shown in Fig.6(e). The corresponding charge densities are indicating in Fig.6(d) and (e), and found that state a/b/c/d exhibits mirror symmetry with respect to state b{}^{\prime }/a{}^{\prime }/d{}^{\prime }/c{}^{\prime }.

Moreover, the smoothness of the strain field can impact the band structure and associated charge density. For N=400, the lowest energy of QVHIS increases when we alter the strain field smoothness from \nu =0 to 0.5, as demonstrated in Fig.7(f). The charge densities at the Fermi energy in Fig.6(f) are are indicated in Fig.7(f), with an expanded penetration depth.

4 Conclusion

The band structure of uniaxial strain superlattice of monolayer and bilayer graphene are systematically investigated. In superlattice where the strain applied along zigzag direction, a controllable and substantial band gap is found, with the maximal band gap achieving 1200/900 meV for monolayer/bilayer graphene. When the strain applied along armchair direction, partial flat bands emerged, leading to induce quantum valley Hall edge state in bilayer graphene with vertical electric field applied. Our finding provides a strategy for manipulating electronic properties of graphene, which present a promising platform in devising graphene-based electronic devices.

References

[1] A. O. Denisov, V. Reckova, S. Cances, M. J. Ruckriegel, M. Masseroni, C. Adam, C. Tong, J. D. Gerber, W. W. Huang, K. Watanabe, T. Taniguchi, T. Ihn, K. Ensslin, and H. Duprez, Spinvalley protected Kramers pair in bilayer graphene, Nat. Nanotechnol. 20(4), 494 (2025)

[2] M. Jugovac, I. Cojocariu, V. Feyer, S. Blügel, G. Bihlmayer, and P. Perna, Spindependent electronic phenomena in heavily-doped monolayer graphene, Carbon 230, 119666 (2024)

[3] P. Blake, E. W. Hill, A. H. Castro Neto, K. S. Novoselov, D. Jiang, R. Yang, T. J. Booth, and A. K. Geim, Making graphene visible, Appl. Phys. Lett. 91(6), 063124 (2007)

[4] C. Casiraghi, A. Hartschuh, E. Lidorikis, H. Qian, H. Harutyunyan, T. Gokus, K. S. Novoselov, and A. C. Ferrari, Rayleigh imaging of graphene and graphene layers, Nano Lett. 7(9), 2711 (2007)

[5] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Electric field effect in atomically thin carbon films, Science 306(5696), 666 (2004)

[6] S. V. Morozov, K. S. Novoselov, M. I. Katsnelson, F. Schedin, D. C. Elias, J. A. Jaszczak, and A. K. Geim, Giant intrinsic carrier mobilities in graphene and its bilayer, Phys. Rev. Lett. 100(1), 016602 (2008)

[7] K. S. Kim, Y. Zhao, H. Jang, S. Y. Lee, J. M. Kim, K. S. Kim, J. H. Ahn, P. Kim, J. Y. Choi, and B. H. Hong, Large-scale pattern growth of graphene films for stretchable transparent electrodes, Nature 457(7230), 706 (2009)

[8] J. Ye, M. F. Craciun, M. Koshino, S. Russo, S. Inoue, H. Yuan, H. Shimotani, A. F. Morpurgo, and Y. Iwasa, Accessing the transport properties of graphene and its multilayers at high carrier density, Proc. Natl. Acad. Sci. USA 108(32), 13002 (2011)

[9] A. K. Geim, Graphene: Status and prospects, Science 324(5934), 1530 (2009)

[10] E. McCann and M. Koshino, The electronic properties of bilayer graphene, Rep. Prog. Phys. 76(5), 056503 (2013)

[11] A. V. Rozhkov, A. O. Sboychakov, A. L. Rakhmanov, and F. Nori, Electronic properties of graphene-based bilayer systems, Phys. Rep. 648, 1 (2016)

[12] G. Yang, L. Li, W. B. Lee, and M. C. Ng, Structure of graphene and its disorders: A review, Sci. Technol. Adv. Mater. 19(1), 613 (2018)

[13] T. Radsar, H. Khalesi, and V. Ghods, Graphene properties andaapplications inananoelectronic, Opt. Quantum Electron. 53(4), 178 (2021)

[14] S. Sahu and G. C. Rout, Band gap opening in graphene: A short theoretical study, Int. Nano Lett. 7(2), 81 (2017)

[15] M. S. Nevius, M. Conrad, F. Wang, A. Celis, M. N. Nair, A. Taleb-Ibrahimi, A. Tejeda, and E. H. Conrad, Semiconducting graphene from highly ordered substrate interactions, Phys. Rev. Lett. 115(13), 136802 (2015)

[16] S. Y. Zhou, G. H. Gweon, A. V. Fedorov, P. N. First, W. A. de Heer, D. H. Lee, F. Guinea, A. H. Castro Neto, and A. Lanzara, Substrate-induced band gap opening in epitaxial graphene, Nat. Mater. 6(10), 770 (2007)

[17] G. Giovannetti, P. A. Khomyakov, G. Brocks, P. J. Kelly, and J. van den Brink, Substrate-induced band gap in graphene on hexagonal boron nitride: Ab initio density functional calculations, Phys. Rev. B 76(7), 073103 (2007)

[18] I. Zanella, S. Guerini, S. B. Fagan, J. Mendes Filho, and A. G. Souza Filho, Chemical doping-induced gap opening and spin polarization in graphene, Phys. Rev. B 77(7), 073404 (2008)

[19] P. P. Shinde and V. Kumar, Direct band gap opening in graphene by BN doping: Ab initio calculations, Phys. Rev. B 84(12), 125401 (2011)

[20] W. J. Yu, L. Liao, S. H. Chae, Y. H. Lee, and X. Duan, Toward tunable band gap and tunable Dirac point in bilayer graphene with molecular doping, Nano Lett. 11(11), 4759 (2011)

[21] X. Fan, Z. Shen, A. Q. Liu, and J. L. Kuo, Band gap opening of graphene by doping small boron nitride domains, Nanoscale 4(6), 2157 (2012)

[22] C. K. Chang, S. Kataria, C. C. Kuo, A. Ganguly, B. Y. Wang, J. Y. Hwang, K. J. Huang, W. H. Yang, S. B. Wang, C. H. Chuang, M. Chen, C. I. Huang, W. F. Pong, K. J. Song, S. J. Chang, J. H. Guo, Y. Tai, M. Tsujimoto, S. Isoda, C. W. Chen, L. C. Chen, and K. H. Chen, Band gap engineering of chemical vapor deposited graphene by in situ BN doping, ACS Nano 7(2), 1333 (2013)

[23] L. Ci, L. Song, C. Jin, D. Jariwala, D. Wu, Y. Li, A. Srivastava, Z. F. Wang, K. Storr, L. Balicas, F. Liu, and P. M. Ajayan, Atomic layers of hybridized boron nitride and graphene domains, Nat. Mater. 9(5), 430 (2010)

[24] M. Y. Han, B. Özyilmaz, Y. Zhang, and P. Kim, Energy band-gap engineering of graphene nanoribbons, Phys. Rev. Lett. 98(20), 206805 (2007)

[25] K. Wakabayashi, M. Fujita, H. Ajiki, and M. Sigrist, Electronic and magnetic properties of nanographite ribbons, Phys. Rev. B 59(12), 8271 (1999)

[26] G. Cocco, E. Cadelano, and L. Colombo, Gap opening in graphene by shear strain, Phys. Rev. B 81(24), 241412 (2010)

[27] N. S. Bandeira, D. R. da Costa, A. Chaves, G. A. Farias, and R. N. C. Filho, Gap opening in graphene nanoribbons by application of simple shear strain and in-plane electric field, J. Phys.: Condens. Matter 33(6), 065503 (2021)

[28] V. M. Pereira, A. H. Castro Neto, and N. M. R. Peres, Tight-binding approach to uniaxial strain in graphene, Phys. Rev. B 80(4), 045401 (2009)

[29] J. Feng, H. Tan, J. F. Liu, and J. Wang, Field-effect tunneling between quantum valley Hall edge states and topological transistors based on bilayer graphene, Phys. Rev. Appl. 19(2), 024072 (2023)

[30] J. Jung, F. Zhang, Z. Qiao, and A. H. MacDonald, Valley-Hall kink and edge states in multilayer graphene, Phys. Rev. B 84(7), 075418 (2011)

[31] Z. Qiao, J. Jung, Q. Niu, and A. H. MacDonald, Electronic highways in bilayer graphene, Nano Lett. 11(8), 3453 (2011)

[32] J. Li, K. Wang, K. J. McFaul, Z. Zern, Y. Ren, K. Watanabe, T. Taniguchi, Z. Qiao, and J. Zhu, Gate-controlled topological conducting channels in bilayer graphene, Nat. Nanotechnol. 11(12), 1060 (2016)

[33] E. V. Castro, N. M. R. Peres, J. M. B. Lopes dos Santos, A. H. C. Neto, and F. Guinea, Localized states at zigzag edges of bilayer graphene, Phys. Rev. Lett. 100(2), 026802 (2008)

[34] F. Zhang, A. H. MacDonald, and E. J. Mele, Valley Chern numbers and boundary modes in gapped bilayer graphene, Proc. Natl. Acad. Sci. USA 110(26), 10546 (2013)

[35] Z. Qiao, W. K. Tse, H. Jiang, Y. Yao, and Q. Niu, Two-dimensional topological insulator state and topological phase transition in bilayer graphene, Phys. Rev. Lett. 107(25), 256801 (2011)

[36] K. W. Lee and C. E. Lee, Half-metallic quantum valley Hall effect in biased zigzag-edge bilayer graphene nanoribbons, Phys. Rev. B 95(8), 085145 (2017)

[37] K. W. Lee and C. E. Lee, Transverse electric field-induced quantum valley Hall effects in zigzag-edge graphene nanoribbons, Phys. Lett. A 382(32), 2137 (2018)

[38] J. Feng, H. Tan, J. F. Liu, and J. Wang, Field-effect tunneling between quantum valley Hall edge states and topological transistors based on bilayer graphene, Phys. Rev. Appl. 19(2), 024072 (2023)

[39] R. Li, P. H. Fu, J. F. Liu, and J. Wang, Armchair edge states in shear-strained graphene: Magnetic properties and quantum valley Hall edge states, Phys. Rev. B 109(4), 045403 (2024)

[40] G. G. Naumis, S. Barraza-Lopez, M. Oliva-Leyva, and H. Terrones, Electronic and optical properties of strained graphene and other strained 2D materials: A review, Rep. Prog. Phys. 80(9), 096501 (2017)

[41] F. Guinea, M. I. Katsnelson, and A. K. Geim, Energy gaps and a zero-field quantum Hall effect in graphene by strain engineering, Nat. Phys. 6(1), 30 (2010)

[42] R. Banerjee, V. H. Nguyen, T. Granzier-Nakajima, L. Pabbi, A. Lherbier, A. R. Binion, J. C. Charlier, M. Terrones, and E. W. Hudson, Strain modulated superlattices in graphene, Nano Lett. 20(5), 3113 (2020)

[43] A. Reserbat-Plantey, D. Kalita, Z. Han, L. Ferlazzo, S. Autier-Laurent, K. Komatsu, C. Li, R. Weil, A. Ralko, L. Marty, S. Guéron, N. Bendiab, H. Bouchiat, and V. Bouchiat, Strain superlattices and macroscale suspension of graphene induced by corrugated substrates, Nano Lett. 14(9), 5044 (2014)

[44] Y. Zhang, Y. Kim, M. J. Gilbert, and N. Mason, Magnetotransport in a strain superlattice of graphene, Appl. Phys. Lett. 115(14), 143508 (2019)

[45] H. Yu and J. F. Liu, Perfect Spin-filtering in graphene monolayer-bilayer superlattice with zigzag boundaries, Sci. Rep. 6(1), 25361 (2016)

[46] R. Li, J. F. Liu, and J. Wang, Strain-modulated perfect valley precession and valley transistor in graphene, Phys. Rev. Appl. 19(2), 024075 (2023)

[47] A. Pachoud, A. Ferreira, B. Özyilmaz, and A. H. Castro Neto, Scattering theory of spin−orbit active adatoms on graphene, Phys. Rev. B 90(3), 035444 (2014)

[48] G. Giovannetti, M. Capone, J. van den Brink, and C. Ortix, Kekulé textures, pseudospin-one Dirac cones, and quadratic band crossings in a graphene-hexagonal indium chalcogenide bilayer, Phys. Rev. B 91(12), 121417 (2015)

[49] J. W. F. Venderbos, M. Manzardo, D. V. Efremov, J. van den Brink, and C. Ortix, Engineering interaction-induced topological insulators in a substrate-induced honeycomb superlattice, Phys. Rev. B 93(4), 045428 (2016)

[50] E. Andrade, R. Carrillo-Bastos, and G. G. Naumis, Valley engineering by strain in Kekulé-distorted graphene, Phys. Rev. B 99(3), 035411 (2019)

[51] H. K. Ke, R. Li, and J. F. Liu, Spin splitting in graphene nanoribbons under non-uniform strain, Results Phys. 56, 107206 (2024)

RIGHTS & PERMISSIONS

Higher Education Press